!AS/Park City Mathematics Series
Volume 12, 2002Langlands-Shahidi Method
Freydoon Shahidi
Foreword
In a series of 8 lectures, titled "Langlands-Shahidi method and Converse The-
orems" Cogdell and Shahidi reviewed these two topics which have played a central
role in recent developments in Langlands Functoriality Conjecture and their num-
ber theoretic consequences. We shall refer the reader to [Cl], [Co], [CPS3], [He2],
[Mi] and [Shll], as well as Sarnak's article in these proceedings and Section 4.2 of
the present notes for several surveys of different aspects of these results.
While Cogdell's lectures [Co] were devoted to the both direct and converse
theory of the Rankin- Selberg product £-functions for GL(n), these notes are the
written version of author's brief expository lectures on Langlands- Shahidi method.
This is a method which exploits the analytic properties of Eisenstein series, their
constant and non-constant terms to derive the analytic properties of automorphic
£-functions which appear in these constant terms. I refer to [GS2] for an earlier
review of this method. A detailed account of the method is the subject matter of
an upcoming book which is in preparation by the author.
Many of the lectures given during the summer school discuss different aspects
of the spectral theory of automorphic forms and are consequently related to differ-
ent parts of the theory discussed here. Notable among them are those of Bernstein,
Borel, Clozel, Cogdell, Michel and Sarnak. For example, while here we have dis-
cussed some of the general techniques used in the proof of the important case of
the transfer of generic cusp forms on classical groups to GL(n) (cf. [CKPSSl,2]),
we have left the discussion of this transfer entirely to Cogdell [Co].
I would like to thank the PCMI staff for their wonderful organization and
support during the three memorable weeks in P art City. Thanks are also due to
Herb Clemens and the entire steering committee of PCMI for allowing me to be
part of the organization of this special event. It has been a pleasure to work with
Peter Sarnak on this.
(^1) Department of Mathematics, Purdue University, West Lafayette, Indiana, USA 47907.
E-mail address: shahidi©math. purdue. edu.
The author was partially supported by NSF grant DMS-0200325 and a Guggenheim fellowship.
@2007 American Mathematical Society
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